The implementation of phase-shifting or heterodyne interferometry (PSI) techniques has become well accepted for interferometric systems that are both commercially available and under development today. These systems are applied to a variety of tasks, including optical testing, surface roughness measurement, distance measurements, and dimensional gauging. The primary reason for the acceptance of the phase-shifting procedures is the extreme precision to which measurements can be made. In fact, the measurement precision of many of these instruments exceeds the calibration standards available. Other important advantages of these phase-shifting techniques over other types of interferometric measurement are the ability to collect data on a regularly spaced grid or at equal time intervals instead of only at fringe centers or zero crossings, and their general immunity to a number of noise sources.
The major limitation of the existing phase-shifting or heterodyne techniques is their inability to measure the surface profile of surfaces with large departures from a best-fit reference surface. With the advent of such techniques as single-point diamond turning, computer controlled polishing, and the molding of aspheric lenses in both plastic and glass, it has become possible to design and fabricate optical surfaces with very large aspheric departures. The existing measurement tools are unable to conveniently, economically and rapidly test the resulting surfaces or wavefronts. The reason for this limitation is that the current phase-shifting algorithm with only correctly reconstruct the wavefront if the change of the wavefront between adjacent measurement points either in space or time is a half wave or less.
Phase-shifting interferometers operate by introducing a time varying phase-shift between the reference and sample beams of an interferometer, which results in a time varying interference pattern. Synchronous detection, either digital or analog, is applied to this pattern to measure the relative phase of the wavefront at an array of sample locations. To simplify the following description, only the phase-shifting technique as applied to spatially varying wavefronts will be described. It should be clear to one skilled in the art however that the techniques described herein will also be readily applied to any of the other detection schemes.
A general expression for the intensity pattern i(x,y,.delta.) of an interferogram is EQU i(x,y,.delta.)=i'(x,y)+i"(x,y) cos [.phi.(x,y)-.delta.], (1)
where .phi.(x,y) is the unknown phase difference between the reference and sample wavefronts, .delta. is the time-varying phase shift introduced between two beams in an interferometer, and i'(x,y) and i"(x,y) are unknown quantities relating to a DC level and a base modulation of the intensity pattern, respectively. The phase shift .delta. is usually obtained by translating one of the optical elements with a piezoelectric drive, although other mechanisms exist. The detection scheme for PSI consists of recording a series of interferograms recorded for different values of the phase .delta.. In practice, the phase shift is often allowed to vary linearly over a range .DELTA. during each of the measurements. The n.sup.th sample interferogram can then be expressed by the integral ##EQU1## where .delta..sub.n is the phase shift at the center of each integration period. Substitution, integration and simplification of equation (2) yields ##EQU2##
The data set needed for analysis of the wavefront using PSI is a set of three or more interferograms recorded with different average phase shifts .delta..sub.n. An example is the four-step algorithm where the phase is advanced in four equal steps of 90.degree.. In this case, EQU .delta..sub.n =0, .pi./2, .pi., 3.pi./2, (4)
and .DELTA. equals zero. The four recorded interferograms after trigonometric simplification are ##EQU3## Combining these equations and solving for .phi.(x,y) gives the result ##EQU4## This last equation is evaluated at every point in the interferogram to yield a map of the wavefront phase. The phase values can easily be converted into optical path differences (OPD's) by: EQU OPD(x,y)=.phi.(x,y).lambda./2.pi. (7)
where .lambda. is the wavelength of the light beam in the interferometer. If the signs of the numerator and denominator of equation (6) are determined, the phase can be calculated over a range of 2.pi.. The result of the arctangent in equation (6) is to give the phase of the wavefront modulo 2.pi.. In order for the data to be useful, any 2.pi. phase discontinuities resulting from the arctangent must be removed. In other words the calculated wavefront returns to a value of zero every time the actual wavefront equals a multiple of 2.pi., and this segmented or "compressed" wavefront must be reconstructed to obtain the correct absolute wavefront. This situation is shown in FIG. 3, where the line labeled W represents the absolute phase of the wavefront, and the segments labeled W' represent the phase of the wavefront modulo 2.pi..
The procedure for removing the 2.pi. phase discontinuities is to start at a single sample value of the wavefront, normally at the center of the interferogram, and to assume that the phase between any two adjacent samples does not change by more than .pi.. If the phase difference calculated for two adjacent samples exceeds .pi., then 2.pi. is added to or subtracted from the value of the second sample to correctly reconstruct the wavefront. The entire wavefront map is then reconstructed by working outward in this manner from the starting sample.
This method of measurement therefore places a restriction on the types of wavefronts and surfaces that can be correctly reconstructed. The condition that the phase of the wavefront changes by less than .pi. per sample is equivalent to restricting the slope of the wavefront to a half wave per sample (or a quarter wave per sample for a surface tested in reflection). Because of this restriction, only surfaces or wavefronts that have small departures from a best-fit reference surface can be tested. For highly aspheric surfaces, the wavefront changes too rapidly for the reconstruction algorithm to keep up with the changes.
Since a complete fringe is formed in the interferogram every time the wavefront changes by a full wave or 2.pi., the maximum fringe frequency that can be measured is equal to half the sampling frequency of the detector recording the interferogram. According to sampling theory, if there are two samples per fringe, then it is always possible to correctly reconstruct the fringe frequency. The PSI algorithm matches this condition. The maximum allowable fringe frequency that can be reconstructed is therefore equal to the Nyquist frequency of the sensor, which is defined to be half the sampling frequency. Frequencies in the interferogram above the Nyquist frequency are aliased to a lower spatial frequency by the sensor, and the PSI reconstruction algorithm is unable to interpret this aliased data.
Because the limitations of PSI are placed on the wavefront slope and not the wavefront itself, it is not possible to predict the number of waves of asphericity that a particular interferometer configuration will be capable of measuring. However available interferometers taking approximately 200 samples across the diameter of an interferogram are typically able to measure aspheric departures of from 10 to 20 waves before the reconstruction algorithm breaks down.
Attempts to overcome this reconstruction have included the use of null lenses or computer generated holograms to reduce the degree of asphericity present in the test wavefront. Detector arrays with increased number of pixels have been used to increase the Nyquist frequency of the sensor. The use of longer wavelengths and two-wavelength techniques, and the use of light sources with limited coherence lengths have also been attempted. While these techniques do increase the measurement range of phase shifting interferometry, and allow the surface to be measured, none of them are entirely satisfying. They all involve a trade-off which either greatly increases the cost of the instrument, places long lead times in the design of the test, makes the instrument much more difficult to calibrate, or decreases the precision of the test.
It is therefore the object of the present invention to provide a technique for extending the measurement range of interferometry that is free from the drawbacks noted above with respect to the prior art.